Hilbert series, Howe duality and branching for classical groups

Abstract

An extension of the Littlewood Restriction Rule is given that covers all pertinent parameters and simplifies to the original under Littlewood's hypotheses. Two formulas are derived for the Gelfand-Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, one in terms of the dual pair index and the other in terms of the highest weight. For a fixed dual pair setting, all the irreducible highest weight representations which occur have the same Gelfand-Kirillov dimension.We define a class of unitary highest weight representations and show that each of these representations, L, has a Hilbert series H L (q) of the form:where R(q) is an explictly given multiple of the Hilbert series of a finite dimensional representation B of a real Lie algebra associated to L. Under this correspondence L → B , the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group.

The Hilbert Series of Measures of Entanglement for 4 Qubits

“…The general element of k is represented by and array [1], b [1], c [1], a [2], b [2], c [2], a [3], b [3], c [3]} representing 4 matrices the ith being…”

“…The actual calculation was carried out using an array of computers that was established with the help of the NSF to do large scale symbolic calculations (the Groebner array). We would like to thank Enright for the formula alluded to above (which is an outgrowth of his joint work with Jeb Willenbring, [3]) and Hanspeter Kraft for numerous discussions, theorems and counterexamples.…”

Stable branching rules for classical symmetric pairs

“…Removing the stability condition for Littlewood's restriction rules is a delicate problem, which was also addressed in [EW1] and [EW2]. Classically, Newell [Ne] presents modification rules to the Littlewood restriction rules to solve the branching problem outside of the stable range (see [Su] and [Ki2]).…”

“…The parameters which fall within this range are said to be in the stable range. These hypotheses are necessary but for certain µ it is possible to weaken them considerably; see [EW1] and [EW2].…”

Lowest weight modules of Sp2n(R)˜ of minimal Gelfand–Kirillov dimension

“…See [3][4][5][7][8][9] for example. In particular, the Gelfand-Kirillov dimensions of irreducible unitary lowest weight modules of G are calculated in [5] and [9]. Both papers use theta correspondences of compact dual pairs, and their methods do not apply to non-unitary representations.…”